Vector Basics

A vector is a quantity that has both direction (where it’s pointing) and magnitude (how large it is). A vector can be written in many formats:

$$\vec{v}=3i+2j-k=[3,2,-1]=\overrightarrow{AB}$$

Each term in a vector refers to a component i.e. the $i$-component of $\vec{v}$ is 3.

#todo merge with Vector

Properties & Operations §

• Vectors can be added and subtracted by applying the operations to the respected components $$[3,2,1] + [8,-1,5] = [11,1,6] = 11i + j + 6k

Geometrically, $\vec{u} - \vec{v}$ is v in the opposite direction + u. * Vectors **cannot be divided or multiplied** with each other. However, they can be multiplied or divided with a scalar * The [dot product](Vector%20Operation) and [cross product](Vector%20Operation) are vector-specific operations * The zero vector is represented as $\vec{0} = [0,0,0,\dots]$ * Similar to geometric lines, vectors can be [Parallel & Perpendicular](#Parallel%20&%20Perpendicular) ### Magnitude The magnitude of a vector refers to it's **geometric length.** It uses the [Pythagorean Theorem](Pythagorean%20Theorem.md).

|\vec{v}| = \sqrt{a^2 + b^2 + c^2 + \dots}

### Parallel & Perpendicular Two vectors are **parallel** if there is some scalar $k$ that can be multiplied with one vector to obtain another.